Dynamic Programming - Subset Sum Problem


Dynamic Programming - Subset Sum Problem
Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum.
Examples: set[] = {3, 34, 4, 12, 5, 2}, sum = 9  Output:  True  //There is a subset (4, 5) with sum 9.  

Let isSubSetSum(int set[], int n, int sum) be the function to find whether there is a subset of set[] with sum equal to sum. n is the number of elements in set[].
The isSubsetSum problem can be divided into two subproblems
…a) Include the last element, recur for n = n-1, sum = sum – set[n-1]
…b) Exclude the last element, recur for n = n-1.
If any of the above the above subproblems return true, then return true.

Following is the recursive formula for isSubsetSum() problem.
isSubsetSum(set, n, sum) = isSubsetSum(set, n-1, sum) ||                              isSubsetSum(arr, n-1, sum-set[n-1])  
Base Cases:  isSubsetSum(set, n, sum) = false, if sum > 0 and n == 0  
isSubsetSum(set, n, sum) = true, if sum == 0 

-- Be careful about the detail:
dp[sum][index] means whether there is match between 0-j-1, not 0-j.
http://www.zrzahid.com/subset-sum-problem-dynamic-programming/
public static boolean isSubSetSum(final int[] set, final int sum) {
    final int m = set.length;
    final boolean[][] ssTable = new boolean[sum + 1][m + 1];
    // base cases: if m == 0 then no solution for any sum
    for (int i = 0; i <= sum; i++) {
        ssTable[i][0] = false;
    }
    // base case: if sum = 0 then there is only one solution for any input set: just take none of each of the items.
    for (int j = 0; j <= m; j++) {
        ssTable[0][j] = true;
    }

    for (int i = 1; i <= sum; i++) {
        for (int j = 1; j <= m; j++) {
            // solutions excluding last element i.e. set[j-1]
            final boolean s1 = ssTable[i][j - 1];
            // solutions including last element i.e. set[j-1]
            final boolean s2 = (i - set[j - 1]) >= 0 ? ssTable[i - set[j - 1]][j - 1] : false;

            ssTable[i][j] = s1 || s2;
        }
    }

    return ssTable[sum][m];
}
bool isSubsetSum(int set[], int n, int sum)
{
    // The value of subset[i][j] will be true if there is a subset of set[0..j-1]
    //  with sum equal to i
    bool subset[sum+1][n+1];
    // If sum is 0, then answer is true
    for (int i = 0; i <= n; i++)
      subset[0][i] = true;
    // If sum is not 0 and set is empty, then answer is false
    for (int i = 1; i <= sum; i++)
      subset[i][0] = false;
     // Fill the subset table in botton up manner
     for (int i = 1; i <= sum; i++)
     {
       for (int j = 1; j <= n; j++)
       {
         subset[i][j] = subset[i][j-1];
         if (i >= set[j-1])
           subset[i][j] = subset[i][j] || subset[i - set[j-1]][j-1];
       }
     }
    /* // uncomment this code to print table
     for (int i = 0; i <= sum; i++)
     {
       for (int j = 0; j <= n; j++)
          printf ("%4d", subset[i][j]);
       printf("\n");
     } */
     return subset[sum][n];
}
Extension:
how many subsets there are in which subset sum is equal X?
How many subsets of A={1,2,3,…,10} have the property that the sum of their elements is ≥28?
Read full article from Dynamic Programming - Subset Sum Problem

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